A Holomorphic Bundle Criterion for Compactness of Pseudoconcave Solvmanifolds

We prove that a pseudoconcave complex homogeneous space of a connected solvable linear algebraic group is necessarily compact. This resolves a central conjecture in the theory, showing that pseudoconcavity characterizes compactness for this large class of solvmanifolds. The proof combines new pluriharmonic obstructions for C∗-bundles with the structure theory of solvable groups, demonstrating that any noncompact solvmanifold admits a nonconstant pluriharmonic function, contradicting pseudoconcavity. Our result unifies and extends all known partial classifications for this class of solvable linear algebraic groups, establishing pseudoconcavity as a definitive geometric property that forces compactness in the solvable setting.